3 - Mathematics and finite element method of thermal ablation therapy

Abstract

The mathematical foundations of the thermal ablation procedures have a noteworthy role in understanding the destruction of the ablation zones by determining the best settings for the ablation process using a specific ablation method to treat a specific organ. Since thermal ablation is based on increasing or decreasing temperatures for tumor ablation, studying the appropriate heat transfer mathematical formulations for animal/human tissues is essential. Generally, the used applicators/probes in the thermal ablation process apply energy to cool or heat the targeted tissues. However, blood perfusion may remove some of this generated energy. Accordingly, the characteristics and location of the targeted tissues have a major effect on the ablation process outcomes. To study the correct way to control the applied temperature and power to completely ablate the targeted tissues, and maximizing the coagulation volumes, the finite element method (FEM) for modeling the ablation procedure including healthy/tumor tissues, and the applicator based on the tissues and material characteristics are designed. Finite element (FE) modeling depends on the mathematical prototype of the thermal process, which happens during the ablation process of tissues. It describes and analyzes the dynamics of the temperature distribution by considering the bioheat transfer in the tissue and/or blood vessel due to the existence of an energy source (generator). In addition, such a model is used to determine the volume of the ablated regions without destroying the nearby healthy tissues and/or blood vessels by defining the optimal input power which increases the maximum temperature in the targeted tissues. This chapter studies in detail most of the mathematical formulas and relations which govern the ablation process, starting from the governing equations of the generated wave in the different ablation systems, including radiofrequency ablation (RFA), microwave ablation (MWA), laser ablation, ultrasound (US) ablation, cryosurgical ablation, and electroporation ablation, then, the heat transfer process is introduced, including Penne’s bioheat equations, Chen and Holmes equations, Weinbaum and Jiji equation, and Weinbaum, Jiji, and Lemons model. Then tissue contraction during the thermal ablation and the first-order Arrhenius rate equation are explained. Finally, the effect of cooling due to the presence of large blood vessels is highlighted, followed by an explanation of the concept of the FEM.